// Run computes β starting witn an initial guess func (o *ReliabFORM) Run(βtrial float64, verbose bool, args ...interface{}) (β float64, μ, σ, x []float64) { // initial random variables β = βtrial nx := len(o.μ) μ = make([]float64, nx) // mean values (equivalent normal value) σ = make([]float64, nx) // deviation values (equivalent normal value) x = make([]float64, nx) // current vector of random variables defining min(β) for i := 0; i < nx; i++ { μ[i] = o.μ[i] σ[i] = o.σ[i] x[i] = o.μ[i] } // lognormal distribution structure var lnd DistLogNormal // has lognormal random variable? haslrv := false for _, found := range o.lrv { if found { haslrv = true break } } // function to compute β with x-constant // gβ(β) = g(μ - β・A・σ) = 0 var err error gβfcn := func(fy, y []float64) error { βtmp := y[0] for i := 0; i < nx; i++ { o.xtmp[i] = μ[i] - βtmp*o.α[i]*σ[i] } fy[0], err = o.gfcn(o.xtmp, args) if err != nil { chk.Panic("cannot compute gfcn(%v):\n%v", o.xtmp, err) } return nil } // derivative of gβ w.r.t β hβfcn := func(dfdy [][]float64, y []float64) error { βtmp := y[0] for i := 0; i < nx; i++ { o.xtmp[i] = μ[i] - βtmp*o.α[i]*σ[i] } err = o.hfcn(o.dgdx, o.xtmp, args) if err != nil { chk.Panic("cannot compute hfcn(%v):\n%v", o.xtmp, err) } dfdy[0][0] = 0 for i := 0; i < nx; i++ { dfdy[0][0] -= o.dgdx[i] * o.α[i] * σ[i] } return nil } // nonlinear solver with y[0] = β // solving: gβ(β) = g(μ - β・A・σ) = 0 var nls num.NlSolver nls.Init(1, gβfcn, nil, hβfcn, true, false, nil) defer nls.Clean() // message if verbose { io.Pf("\n%s", io.StrThickLine(60)) } // plotting plot := o.PlotFnk != "" if nx != 2 { plot = false } if plot { if o.PlotNp < 3 { o.PlotNp = 41 } var umin, umax, vmin, vmax float64 if o.PlotCf < 1 { o.PlotCf = 2 } if len(o.PlotUrange) == 0 { umin, umax = μ[0]-o.PlotCf*μ[0], μ[0]+o.PlotCf*μ[0] vmin, vmax = μ[1]-o.PlotCf*μ[1], μ[1]+o.PlotCf*μ[1] } else { chk.IntAssert(len(o.PlotUrange), 2) chk.IntAssert(len(o.PlotVrange), 2) umin, umax = o.PlotUrange[0], o.PlotUrange[1] vmin, vmax = o.PlotVrange[0], o.PlotVrange[1] } o.PlotU, o.PlotV = utl.MeshGrid2D(umin, umax, vmin, vmax, o.PlotNp, o.PlotNp) o.PlotZ = la.MatAlloc(o.PlotNp, o.PlotNp) plt.SetForEps(0.8, 300) for i := 0; i < o.PlotNp; i++ { for j := 0; j < o.PlotNp; j++ { o.xtmp[0] = o.PlotU[i][j] o.xtmp[1] = o.PlotV[i][j] o.PlotZ[i][j], err = o.gfcn(o.xtmp, args) if err != nil { chk.Panic("cannot compute gfcn(%v):\n%v", x, err) } } } plt.Contour(o.PlotU, o.PlotV, o.PlotZ, "") plt.ContourSimple(o.PlotU, o.PlotV, o.PlotZ, true, 8, "levels=[0], colors=['yellow']") plt.PlotOne(x[0], x[1], "'ro', label='initial'") } // iterations to find β var dat VarData B := []float64{β} itB := 0 for itB = 0; itB < o.NmaxItB; itB++ { // message if verbose { gx, err := o.gfcn(x, args) if err != nil { chk.Panic("cannot compute gfcn(%v):\n%v", x, err) } io.Pf("%s itB=%d β=%g g=%g\n", io.StrThinLine(60), itB, β, gx) } // plot if plot { plt.PlotOne(x[0], x[1], "'r.'") } // compute direction cosines itA := 0 for itA = 0; itA < o.NmaxItA; itA++ { // has lognormal random variable (lrv) if haslrv { // find equivalent normal mean and std deviation for lognormal variables for i := 0; i < nx; i++ { if o.lrv[i] { // set distribution dat.M, dat.S = o.μ[i], o.σ[i] lnd.Init(&dat) // update μ and σ fx := lnd.Pdf(x[i]) Φinvx := (math.Log(x[i]) - lnd.M) / lnd.S φx := math.Exp(-Φinvx*Φinvx/2.0) / math.Sqrt2 / math.SqrtPi σ[i] = φx / fx μ[i] = x[i] - Φinvx*σ[i] } } } // compute direction cosines err = o.hfcn(o.dgdx, x, args) if err != nil { chk.Panic("cannot compute hfcn(%v):\n%v", x, err) } den := 0.0 for i := 0; i < nx; i++ { den += math.Pow(o.dgdx[i]*σ[i], 2.0) } den = math.Sqrt(den) αerr := 0.0 // difference on α for i := 0; i < nx; i++ { αnew := o.dgdx[i] * σ[i] / den αerr += math.Pow(αnew-o.α[i], 2.0) o.α[i] = αnew } αerr = math.Sqrt(αerr) // message if verbose { io.Pf(" itA=%d\n", itA) io.Pf("%12s%12s%12s%12s\n", "x", "μ", "σ", "α") for i := 0; i < nx; i++ { io.Pf("%12.3f%12.3f%12.3f%12.3f\n", x[i], μ[i], σ[i], o.α[i]) } } // update x-star for i := 0; i < nx; i++ { x[i] = μ[i] - β*o.α[i]*σ[i] } // check convergence on α if itA > 1 && αerr < o.TolA { if verbose { io.Pfgrey(". . . converged on α with αerr=%g . . .\n", αerr) } break } } // failed to converge on α if itA == o.NmaxItA { chk.Panic("failed to convege on α") } // compute new β B[0] = β nls.Solve(B, o.NlsSilent) βerr := math.Abs(B[0] - β) β = B[0] if o.NlsCheckJ { nls.CheckJ(B, o.NlsCheckJtol, true, false) } // update x-star for i := 0; i < nx; i++ { x[i] = μ[i] - β*o.α[i]*σ[i] } // check convergence on β if βerr < o.TolB { if verbose { io.Pfgrey2(". . . converged on β with βerr=%g . . .\n", βerr) } break } } // failed to converge on β if itB == o.NmaxItB { chk.Panic("failed to converge on β") } // message if verbose { gx, err := o.gfcn(x, args) if err != nil { chk.Panic("cannot compute gfcn(%v):\n%v", x, err) } io.Pfgreen("x = %v\n", x) io.Pfgreen("g = %v\n", gx) io.PfGreen("β = %v\n", β) } // plot if plot { plt.Gll("$x_0$", "$x_1$", "") plt.Cross("") plt.SaveD("/tmp/gosl", "fig_form_"+o.PlotFnk+".eps") } return }